A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

Abstract

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.

Keywords:

• Cubic spline method
• uniform mesh discretization
• Euler-Bernoulli beam model
• single and double-soliton of good Boussinesq equation
• Benjamin-Ono equation

MSC Classifications:

• 65M06
• 65M12
• 65M22
• 65Y20

Acknowledgments

The authors are grateful to the handling editor and anonymous referees for their careful reading and valuable comments which have improved the presentation and contents of this paper significantly.

Disclosure statement

The authors declare that they have no conflict of interest/competing interests.